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Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. [3] [4] Computing matrix products is a central operation in all computational applications of linear algebra.
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
The name "dot product" is derived from the dot operator " · " that is often used to designate this operation; [1] the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
The outer product contrasts with: The dot product (a special case of "inner product"), which takes a pair of coordinate vectors as input and produces a scalar; The Kronecker product, which takes a pair of matrices as input and produces a block matrix; Standard matrix multiplication
Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix. The dot product of two column vectors a, b, considered as elements of a coordinate space, is equal to the matrix product of the transpose of a with b,
A common variation of gemm is the gemm3m, which calculates a complex product using "three real matrix multiplications and five real matrix additions instead of the conventional four real matrix multiplications and two real matrix additions", an algorithm similar to Strassen algorithm first described by Peter Ungar. [24]
The simple product of two triple products (or the square of a triple product), may be expanded in terms of dot products: [1] (()) (()) = [] This restates in vector notation that the product of the determinants of two 3×3 matrices equals the determinant of their matrix product.
Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic.